Theorem ddeval0 30298

Description: Value of the delta measure. (Contributed by Thierry Arnoux, 14-Sep-2018.)

Assertion

Ref	Expression
ddeval0	|- ( ( A C_ RR /\ -. 0 e. A ) -> (δ ` A ) = 0 )

Proof of Theorem ddeval0

Dummy variable a is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reex 10027	. . . . 5 |- RR e. _V
2	1	ssex 4802	. . . 4 |- ( A C_ RR -> A e. _V )
3		elpwg 4166	. . . . 5 |- ( A e. _V -> ( A e. ~P RR <-> A C_ RR ) )
4	3	biimpar 502	. . . 4 |- ( ( A e. _V /\ A C_ RR ) -> A e. ~P RR )
5	2, 4	mpancom 703	. . 3 |- ( A C_ RR -> A e. ~P RR )
6		eleq2 2690	. . . . 5 |- ( a = A -> ( 0 e. a <-> 0 e. A ) )
7	6	ifbid 4108	. . . 4 |- ( a = A -> if ( 0 e. a , 1 , 0 ) = if ( 0 e. A , 1 , 0 ) )
8		df-dde 30296	. . . 4 |- δ = ( a e. ~P RR |-> if ( 0 e. a , 1 , 0 ) )
9		1ex 10035	. . . . 5 |- 1 e. _V
10		c0ex 10034	. . . . 5 |- 0 e. _V
11	9, 10	ifex 4156	. . . 4 |- if ( 0 e. A , 1 , 0 ) e. _V
12	7, 8, 11	fvmpt 6282	. . 3 |- ( A e. ~P RR -> (δ ` A ) = if ( 0 e. A , 1 , 0 ) )
13	5, 12	syl 17	. 2 |- ( A C_ RR -> (δ ` A ) = if ( 0 e. A , 1 , 0 ) )
14		iffalse 4095	. 2 |- ( -. 0 e. A -> if ( 0 e. A , 1 , 0 ) = 0 )
15	13, 14	sylan9eq 2676	1 |- ( ( A C_ RR /\ -. 0 e. A ) -> (δ ` A ) = 0 )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ifcif 4086   ~Pcpw 4158   ` cfv 5888   RRcr 9935   0cc0 9936   1c1 9937  δcdde 30295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-mulcl 9998  ax-i2m1 10004

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-dde 30296

This theorem is referenced by:  ddemeas  30299